Homotopical algebra and higher categories

Transcription

1 Homotopical algebra and higher categories Winter term 2016/17 Christoph Schweigert Hamburg University Department of Mathematics Section Algebra and Number Theory and Center for Mathematical Physics (as of: ) Contents 1 Simplicial sets and Kan complexes Basic definitions The Kan Property Connections to Topology Geometric realization Simplicial homotopy Homotopy Groups The Hurewicz Homomorphism Dold-Kan correspondence and Eilenberg-Mac Lane-Spaces Infinity-categories and model categories Basics on -categories Model categories Simplicial categories Categorical constructions with -categories Functors Join construction Slice construction Final and initial objects Limits and colimits The current version of these notes can be found under as a pdf file. 1

2 Please send comments and corrections to These notes are based on lectures delivered at the University of Hamburg in the winter term 2016/17. I am grateful to Jan Hesse, Vincent Koppen, Anssi Lahtinen, Lukas Müller and Lukas Woike for helpful comments. 2

3 1 Simplicial sets and Kan complexes 1.1 Basic definitions We start with the following basic definition: Definition [Simplicial set] A simplicial set K is a sequence of disjoint sets (K n ) n N0, together with maps d i = d i,n : K n K n 1 as well as s i = s i,n : K n K n+1, for 0 i n such that for all 0 j n the following relations hold: d i d j = d j 1 d i if i < j s i s j = s j+1 s i if i j d i s j = s j 1 d i if i < j d i s i = id Kn = d i+1 s i d i s j = s j d i 1 if i > j + 1. (1) The maps d i are called face maps and the maps s i are called degeneracy maps. We complement this definition with several related notions: Definition Let K be a simplicial set K. 1. We call an element x K n an n-simplex of K. An element of K 0 is called a vertex of K. A simplex of the form y = s i x is called degenerate. Otherwise, a simplex is called non-degenerate. 2. Given x K n, we denote the (n + 1)-tuple of faces of x by x := (d 0 x,..., d n x) (K n 1 ) n A subcomplex L of a simplicial set K is a sequence of subsets L n K n that forms, together with the restrictions of the face maps and degeneracy maps, a simplicial set. 4. A simplicial pair (K, L ) is a simplicial set K, together with a subcomplex L K. 5. A pointed simplicial set (K, ) is a simplicial set with a distinguished base point K A pointed simplicial pair (K, L, ) is a simplicial pair, together with a base point L 0. Example A trivial example of a simplicial set is given by taking X n = to be the one-point space in all degrees. Then face and degeneracy maps are unique and the relations trivially hold. For any k N 0, denote by [k] := ({0,..., k}, ) the totally ordered set of the k + 1 elements 0, 1,..., k. 1

4 We use these sets to construct for each fixed n N 0 a simplicial set Δ n, called the standard n-simplex Δ n. The k-th set constituting the simplical set Δ n is the set of order preserving maps to [n]: Δ n k := {x : [k] [n] x non-decreasing} = {(x 0,..., x k ) [n] k+1 0 x 0... x k n} Put differently, Δ n k is the set( of non-decreasing ) sequences of length k with values in the set n + k + 1 [n]. This is a finite set with elements. The face and degeneracy operators n given by omitting and repeating entries: d i (x 0,..., x k ) = (x 0,..., x i 1, x i+1,..., x k ) and s i (x 0,..., x k ) = (x 0,..., x i, x i,...,, x k ). The form of s i (x 0,..., x n ) explains the word degenerate simplex. We also need maps of simplicial sets. Definition A map f between simplicial sets K, L is a sequence of maps (f n : K n L n ) n N0 that is compatible with the face and degeneracy operators: f n 1 d i = d i f n and f n+1 s i = s i f n. 2. Maps of pointed simplicial sets or simplicial pairs f (K, L ) (K, L ) are required to preserve this extra structure, i.e. f( ) = and f(l ) L. Remark In order to specify a simplicial map, it is enough to specify it on non-degenerate simplices. On degenerate simplices, it is then fixed by the relation f(s i (x)) = s i (f(x)). As an example that simplical sets behave nicely, we present the following definition: Definition The product of two simplical sets K and L is defined by (K L) n := K n L n, with the face and degeneracy operators: d i (x, y) = (d i x, d i y) and s i (x, y) = (s i x, s i y) for (x, y) K n L n and i [n]. 2. The coproduct or disjoint union is given by (K L) n := K n L n with operators such that K, L are subcomplexes. Arbitrary (co-)products are defined in a similar manner. This language is very concrete; it is helpful to rephrase our notions in a more abstract way. Definition A category C consists (a) of a class of objects Obj(C), whose entries are called the objects of the category. (b) a class Hom(C), whose entries are called morphisms of the category (c) Maps id : Obj(C) Hom (C) s, t : Hom(C) Obj(C) o : Hom(C) Obj (C) Hom(C) Hom(C) 2

5 such that (a) s(id V ) = t(id V ) = V (b) id t(f) f = f id s(f) = f for all V Obj(C) for all f Hom(C) (c) for all f, g, h Hom(C) with t(f) = s(g) and t(g) = s(h) the associativity identity (h g) f = h (g f) holds. 2. We write for V, W Obj(C) Hom C (V, W ) = {f Hom(C) s(f) = V, t(f) = W } and End C (V ) for Hom C (V, V ). For any pair V, W, we require Hom C (V, W ) to be a set. Elements of End C (V ) are called endomorphisms of V. 3. A morphism f Hom(V, W ) which we also write V f W or in the form f : V W is called an isomorphism, if there exists a morphism g : W V, such that g f = id V and f g = id W. Two objects V, W of a category are called isomorphic, if there is an isomorphism V W. Being isomorphic is an equivalence relation; the equivalence classes of the category C are denoted by π 0 (C). Examples Any set X can be endowed with a trivial structure of a category X in which the only morphisms are the identity morphisms. 2. Vector spaces over a field K, together with linear maps, form a category vect(k). More generally, left modules over a ring R form a category R-mod. K-linear representations of a given group G, together with intertwiners, form a category. 3. Consider a category with a single object ; this category is completely described by the set End( ) which has the structure of an (associative, unital) monoid. 4. A category in which all morphisms are isomorphisms is a called a group oid. A groupoid with single object is completely described by the monoid G := End( ) which is a group. We write //G for this groupoid. 5. An important example of a groupoid is the fundamental groupoid Π 1 (M) of a topological space M: its objects are the points of the space M, a morphism from p M to q M is a homotopy class of paths from p to q. For this groupoid End(x) =: π 1 (X, x) is the fundamental group for the base point x X. The isomorphism classes of Π 1 (M) are the path-connected components of M. 6. From definition 1.1.4, we get categories of simplical sets sset, pointed simplicial sets sset +, simplicial pairs spair and pointed simplicial pairs spair The category of finite ordinals Δ has as objects the totally ordered sets [n] from example and as morphisms order preserving maps. We discuss some examples of morphisms in the category Δ: The coface maps d i : [n 1] [n] for 0 i n 3

6 with d i (j) = j for 0 j i 1 and d i (j) = j +1 for i j n are strictly monotonuously increasing and omit the value i. The codegeneracy maps s i : [n + 1] [n] for 0 i n with s i (i) = s i (i + 1) = i are strictly monotonously increasing, except for taking twice the value i. We find the relations d j d i = d i d j 1 if i < j s j s i = s i s j+1 if i j s j d i = d i s j 1 if i < j s i d i = id = s i d i+1 s j d i = d i 1 s j if i > j + 1. (2) One should compare these relations to the relations (1) and pay attention to the order of the composition. These maps can be seen as a set of generators for the morphisms of Δ in the sense that any morphism in Δ is a composition of these morphisms. More precisely: Any strictly increasing map is a composition of coface maps. Any non-decreasing surjection is a composition of codegeneracy maps. Any non-decreasing map is a composition of coface and codegeneracy maps. The relations given can be seen to be the only relations in Δ. 8. More generally, any partially ordered set (M, ) can be seen as a category. The objects are the elements of M. There is a single morphism n m, iff n m. In particular, each ordinal number [n] by itself can be seen as a category. Definition Let C and C be categories. A functor F : C C consists of two maps: which obey the following conditions: F : Obj(C) Obj(C ) F : Hom(C) Hom(C ), (a) F (id V ) = id F (V ) for all objects V Obj(C) (b) s(f (f)) = F s(f) and t(f (f)) = F t(f) for all morphisms f Hom (C) (c) For any pair f, g of composable morphisms, we have Two functors F (g f) = F (g) F (f). F : C C G : C C can be concatenated to a functor G F : C C by concatenating the maps on objects and morphisms. This concatenation is associative. We discuss some examples of functors: 4

7 Examples A functor //G vect(k) assigns to the single object a K-vector space M and to any group element g G an endomorphism ρ(g) of M. Since functors preserve composition, the map ρ defines a representation of the group G. Thus K-linear representations of G are in bijection to functors //G vect(k). 2. Associating to a vector space V its dual space provides a functor vect(k) vect(k) opp V V. Here we have introduced the opposed category C opp of a category C. It has the same objects as C, but Hom opp (U, V ) := Hom(V, U). The composition is defined in a compatible way. This definition implements the idea of reversing arrows. The bidual provides a functor Bi : vect(k) vect(k) V V. 3. Given two partially ordered sets (M, ) and (N, ), functors f : M N of the corresponding categories are in bijection to order preserving maps. 4. A simplical set as in definition is in bijection to a functor X : Δ opp Set from the opposed category of finite ordinals from example to the category of sets. (One can also say that a simplicial set is a presheaf on Δ.) To see this, consider the sets X n = X([n]). Then d k := X(d k ) are face maps, and s k := X(s k ) degeneracy maps, because the relations (1) and (2) are related by reversing the order of composition. 5. In particular, for any ordinal [n], the functor Hom Δ (, [n]) : Δ opp Set [m] Hom Δ ([m], [n]) defines a simplicial set. This is just the simplical set Δ n introduced in example We now can also talk about simplicial objects for any category C, e.g. there are simplicial manifolds, simplicial groups etc. Definition Let C be any category. A simplicial C-object is a functor Δ opp C. Since coface and codegeneracy maps form a set of generators for the morphisms in Δ, it suffices to specify the corresponding maps and check the relations (1) in definition to specify a simplicial object. Our goal is to show that simplicial sets relate to topological spaces and to categories. We first relate topological spaces to simplicial sets. Example The topological standard simplex is the space Δ n := {(t 0, t 1,..., t n ) R n+1 n t i = 1, t i 0} with the usual subspace topology. This is the convex hull of the vectors e i = (0,... 1, ) of the standard basis of R n+1. 5 i=0

8 We construct a functor Δ Top from the category of ordinals Δ to topological spaces; on objects, we have [n] Δ n. For an order-preserving map θ : [n] [m], we define a map of topological spaces by with θ : Δ n Δ m θ (t 0,..., t n ) = (s 0,..., s m ) s i := j θ 1 (i) where the sum over the empty set is considered as zero. As an example, consider We obtain d i : [n 1] [n] (0, 1,..., n 1) (0, 1,..., i 1, i + 1,..., n 1) (d i ) : Δ n 1 Δ n (t 0,..., t n 1 ) (t 0,..., t i 1, 0, t i,..., t n 1 ) since i is not in the image of d i. The image of (d i ) is the face of the topological simplex that is opposite to the i-th vertex. One should check that this defines a functor Δ Top. Now let T be any topological space. Consider the simplicial set, i.e. the functor Δ opp Set given on objects by [n] Hom Top ( Δ n, T ) and on morphisms by mapping θ : [n] [m] to θ : Hom Top ( Δ m, T ) Hom Top ( Δ n, T ) ϕ ϕ θ This is the simplicial set S(T ), the so-called simplicial singular complex, that gives standard singular homology of the topological space T. It knows all about the topological space we wish to know in algebraic topology. Remark We can now perform for simplicial sets algebraic constructions we know from algebraic topology: 1. Given a simplicial set Y : Δ opp Set, we construct a simplicial abelain group, i.e. a functor Δ opp Ab by composing with the functor F : Set Ab that assigns to any set M the free abelian group F (M) on M. t j We thus obtain the simplicial abelian group ZY : Δ opp being the free abelian group on the set Y n. Y Set F Ab. In particular, ZY n 6

9 2. Morphisms in an abelian group can be added and subtracted. We use this to associate to a simplicial abelian group A its Moore complex A 0 A 1 A This is the complex of abelian groups with the group A i in degree i and differential = n ( 1) i d i i=0 in degree n. The simplicial relations (1) imply as in [AT-SS16, Lemma ] that this is a complex. By abuse of language, the Moore complex obtained from a simplicial set A via the simplicial group Z A is again denoted by ZA. 3. The singular homology groups H (T, A) of a topological space T with coefficients in an abelian group A are the homology groups of the complex ZS(T ) A. We can now talk about the homology groups of a simplicial set S with coefficients in an abelian group A as the homology groups H (ZS A). Categories yield simplicial sets: given a small category C, one can form the simplicial set N(C) sset, called the nerve of C. Example By definition, we have N(C) n = Fun([n], C), where the ordinal [n] is considered as a category and Fun(, ) is the set of functors. Explicitly, an element in N(C) n is a functor F : [n] C. It is given on objects by an (n + 1)-tuple (X 0, X 1,..., X n ) of objects X i = F (i) and morphisms f i := F (i i + 1). These morphisms are thus composable and N(C) n is the set of strings of n composable morphisms in C. Note that N(C) 0 is the set of objects of C. 2. The face maps d i : N(C) n N(C) n 1 are given for i = 1,... n 1 by composing the i 1-th and i-th morphism. The face map d 0 drops the first morphism, the face map d n the last morphism. The degeneracy map s i : N(C) n N(C) n+1 is given by inserting the identity at the i-th position. 3. Consider the ordinal [n] as a category. Its nerve is the simplicial set Δ n = Hom Δ (, [n]) we constructed in, represented by [n] Δ. Remark Suppose that we know that the simplicial set S is the nerve of a category. We then obtain the set of objects as S 0 and the set of morphisms as S 1. We have two face maps d 0, d 1 : S 1 S 0 which give the source and the target of a morphism. The identity morphism on x S 0 is given by the degeneracy map s 0 (x) S Note that the simplicial relation d 0 s 0 = d 1 s 1 from (1) implies that source and target of s 0 (x) are identical. 3. In the definition of the nerve, N(C) 2 consists of functors defined on the category

10 This picture makes clear that the first face map d 1 : N(C) 2 N(C) 1 is essentially given by composition, and that, strictly speaking, a 2-simplex in N(C) is a pair of composable morphisms, together with their composition. In this way, the nerve also contains the information about the composition in the category. Example Applying the nerve construction to the groupoid //G with G a group gives an example of a simplicial group BG: we have BG n = G n, with G 0 the trivial group. For an element (g 1,..., g n ) BG n, we have d 0 (g 1,..., g n ) = (g 2,..., g n ) d i (g 1,..., g n ) = (g 1, g 2,..., g i g i+1,... g n ) for i = 1,..., n 1 d n (g 1,..., g n ) = (g 1, g 2,..., g n 1 ) s i (g 1,..., g n ) = (g 1, g 2,..., g i, e, g i+1,... g n ) This simplicial set is crucial for group cohomology. One should check that a group homomorphism f : G H induces a map Bf : BG BH of simplicial sets. We have now associated to a topological space a simplicial set and to a category the nerve as a simplicial set. This is a first hint that simplicial sets provide a unifying framework for categories and topological spaces. Indeed, they will provide a framework for a generalization of categories, -categories. It is important to compare two functors F, G : C C between the same categories. We give three motivations: We have seen in example that for G a group, a functor F ρ : //G vect(k) corresponds to a K-linear representation of the group G. We know that there are intertwiners between different representations. Given two functors F ρ, F ρ : //G vect(k), we thus need notion of a morphism of functors. A single simplical set is a functor X : Δ opp Set. We need to capture morphisms of simplicial sets to capture the category sset from defintion To get an idea on how to relate functors, we remark that any vector space V can be embedded into its bidual vector space. This means that for every V there is a linear map ι V : id(v ) = V V = Bi(V ) v (β β(v)) that relates the two functors id, Bi : vect(k) vect(k). We formalize this as follows: Definition Let F, G : C C be functors. A natural transformation is a family of morphisms η : F G η V : F (V ) G(V ) in C, indexed by objects V Obj(C) in the source category such that for any morphism f : V W 8

11 in the source category C the diagram in C F (V ) F (f) F (W ) η V G(V ) G(f) η W G(W ) commutes. 2. If for each object V Obj(C) the morphism η V is an isomorphism, then η : F G is called a natural isomorphism. 3. A functor F : C D is called an equivalence of categories, if there is a functor G : D C and natural isomorphisms η : id D F G θ : GF id C. Example Let G be a finite group, K a field and consider two functors F ρ, F ρ : //G vect(k). A natural transformation η : F ρ F ρ is a K-linear map η : F ρ ( ) F ρ ( ) which by the commuting diagram in is an intertwiner of G-representations. 2. If the class Obj(C) is a set, then there is a category Fun(C, C ) whose objects are functors F, G : C C and whose morphisms natural transformations η : F G. In this way, we obtain the category of simplicial sets as sset = Fun(Δ, Set) and, more general, categories Fun(Δ, C) of simplicial C-objects. Definition A category C is called small,, if its class of objects Obj(C) is a set. A category C is called essentially small,, if it is equivalent to a small category. The following lemma is useful to find equivalences of categories: Lemma A functor F : C D is an equivalence of categories, if and only if the following two conditions hold: (a) The functor F is essentially surjective, i.e. for any W Obj(D) there is V Obj(C) such that F (V ) = W in D. (b) The functor F is fully faithful: for any pair V, V of objects in C, the map on Hom-spaces is bijective. F : Hom C (V, V ) Hom D (F (V ), F (V )) 9

12 Proof: see [Kassel, p. 278]. The proof uses the axiom of choice. Observation Consider the category Cat whose objects are categories and whose morphisms are functors. The nerve assigns to each object C Cat a simplicial set N(C) sset. From a functor F : C C, we can construct a map of simplicial sets which is given in degree n by N(F ) n : N(C) n = Fun([n], C) N(C ) n = Fun([n], C ) ϕ F ϕ. It is readily checked that this turns N into a functor N : Cat sset. We characterize its image. To this end, we need the notion of a pullback. Definition Let C be any category. Suppose we are given a hook diagram Γ := X f Y g M. A pullback of Γ is an object Z with morphisms Z X and Z Y such that the square Z Y g X f M commutes and such that for any object W with morphisms W X and W Y such that the analogous square commutes, there is a unique morphism W Z: W! Z Y X M We write Z = X M Y = X f g Y and call Z the pullback of X M along Y M. The diagram Z X is called a pullback diagram. Y g f M Remarks Since pullbacks are defined by a universal property, they are unique up to unique isomorphism. 10

13 2. In the categories Set, Top, Ab, R Mod, the pullback is given by a subobject of the product, X M Y = X f g Y = {(x, y) x X, y Y, f(x) = g(y)}. Structure morphisms are restrictions of projections. The morphism Z X M Y induced by two suitable morphisms ϕ : Z X und ψ : Z Y is just z (ϕ(z), ψ(z)). In the category Top of topological spaces, we use an initial topology. The pullback is an example of a difference kernel: for two morphisms f, g : N M, the difference kernel is a morphism Eq N such that in the diagram Eq N f M g h! h X there exists for any morphism h : X N satisfying f h = g h a unique morphism h : X Eq such that the diagram commutes. In an additive category, the difference kernel is the kernel of the difference map δ(x, y) := f(x) g(y). Thus, in the categories Set, T op, Ab, R-mod, the pullback is the difference kernel f pr 1 X M Y = X f g Y = Eq( X Y M ). g pr 2 3. If in the diagram X 2 X 1 X 0 M 2 M 1 M 0 the two smaller diagrams are pullback diagrams, then also the big diagram is a pullback diagram. We thus have To see this, consider the diagram (X 0 M0 M 1 ) M1 M 2 = X0 M0 M 2. Y (X 0 M0 M 1 ) M1 M 2 X 0 M0 M 1 X 0 f 1 M f 0 2 M 1 M 0 for any object Y. Consider morphisms Y X 0 and Y M 2 such that the outermost rectangle commutes. The pullback property of the right diagram, applied to the morphisms Y X 0 and Y M 2 f 1 M1 yields a unique morphism Y X 0 M0 M 1. Applying to this morphism and to Y M 2 the pullback property of the left diagram, we get a unique morphism Y (X 0 M0 M 1 ) M1 M 2, which shows that the big diagram is a pullback diagram as well. 11

14 Definition Dually to the pullback, we have the pushout of a diagram of the form as a diagram N f A g B N f A g B W written as W = A N B, with the dual universal property. Remark In the category R-mod, pushout is given by the quotient module A N B = A B/(f(n), 0) (0, g(n)). The structure morphisms A, B A N B are given by the injection of the direct sum, followed by the canonical projection to the quotient. The morphism A N B Z induced by two suitable morphisms f : A Z and g : B Z is given by [a, b] f(a) + g(b). It is well-defined on the quotient. More generally, the pushout is the difference cokernel of the map A N B := CoEq( N f g A B ) into the coproduct, provided the coproduct and the cokernel exist. It is a quotient of a coproduct. In the category Top of topological spaces, we use a final topology. 2. All statements about the pullback have obvious analogues for the pushout. Observation Given a non-decreasing sequence 0 i 0 i 1... i m n, we get a unique morphism p i0,i 1,...,i m : [m] [n] with the image given by the non-decreasing sequence. This gives a map of sets p i 0,i 1,...,i m : N(C) n = Fun([n], C) N(C) m = Fun([m], C) ϕ ϕ p i0,i 1,...,i m The equality of maps p 12 p 0 = p 01 p 1 implies a commuting diagram For the nerve of a category, we have pullback diagrams of sets N(C) 2 p 01 p 12 N(C) 1 p 0 N(C) 1 p 1 N(C) 0 12

15 For the nerve of a category, this is a pullback diagram of sets which determines an isomorphism of sets q : N(C) 2 N(C) 1 N0 (C) N(C) 1. A pair of composable morphisms is an element (f, g) N(C) 1 N0 (C) N(C) 1 and composition of f and g can be recovered as g f = p 0,2q 1 (f, g). In a general simplicial set, we just have commuting diagrams X 2 p 01 p 12 X 1 p 0 X 1 p 1 X 0 which are not pulback diagrams and induce a map q : X 2 X 1 X0 X 1 that is not necessarily an isomorphism. Definition [Segal condition] A simplicial object X sset obeys the Segal condition, if for any pair of integers m, n the diagram p X 0,1,...m m+n X m p m,m+1,...,m+n X n p 0 p m X 0 is a pullback square. Proposition (Nerve theorem). A simplicial set X sset is the nerve N(C) of a category C, if and only if it obeys the Segal condition, i.e. if for all k and families (n i ) i=1,...k, the canonical map is an isomorphism. X n1 + +n k X n1 X0 X n2 X0 X0 X nk Proof. We only give the idea: It is clear from the definition of composable morphisms that the nerve of a category obeys the Segal condition. Conversely, the Segal condition allows to express X n = X 1 X0 X 1 X0... X0 X 1 in terms of X 1 and X 0. We can now finally state: Lemma The nerve functor N : Cat sset is fully faithful and hence induces an equivalence onto its essential image. Proof. 13

16 A functor is characterized by its maps on objects and on morphisms. Given a functor F : C C, these maps can be recovered from the components N(F ) 0 : N(C) 0 = Obj(C) N(C ) 0 = Obj(C ) N(F ) 1 : N(C) 1 = Hom(C) N(C ) 0 = Hom(C ) Hence, the nerve functor N is injective on morphisms. We have to show that any map f : N(C) N(C ) of simplicial sets comes from a functor F : C C. The functor consists of a map on objects and morphisms. It is clear that these maps should be given by f 0 and f 1. The simplicial relations imply that this defines a functor. The Segal condition ensures that then the map is determined by the functor everywhere. Hence we are able to characterize the essential image of the nerve functor: Proposition Via the nerve functor, the category Cat is equivalent to the full subcategory of simplicial sets obeying the Segal condition. Observation Let C be a category. Any object c C gives rise to a functor h c = Hom C (c, ) : C Set sending X C to the set Hom(c, X) and the morphism X f Y in C to the map of sets Note that there is also a functor Hom C (c, X) Hom C (c, Y ) ϕ f ϕ h c = Hom C (, c) : C opp Set c Hom C (c, c) sending the morphism X f Y in C to the map of sets Hom C (Y, c) Hom C (X, c) ϕ ϕ f If C is (essentially) small so that Fun(C, Set) is a category, this gives a functor h : C opp Fun(C, Set). Definition A functor F : C opp Set that is isomorphic to a functor of the form h c called representable. It is said to be represented by the object c C. 14

17 Proposition (Yoneda lemma). Let C be a category and F : C Set be any functor. For any object c C, we have an isomorphism Nat(h c, F ) F (c) α α c (id c ). This isomorphism is natural in c and F when both sides are regarded as functors from Fun(C, Set) C to Set. Note that we have so that indeed α c (c) F (c). α c : Hom(c, c) F (c) Proof. If N : h c F is a natural transformation, then for any morphism f : c c in C the diagram in the category Set h c (c) hc (f) h c (c ) N c N c F (c) F (f) F (c ) commutes. Chasing the image of id c h c (c) = Hom C (c, c) through the diagram, we find using that the equality N c (h c (f)) = N c (f id c ) = N c (f) N c (f) = F (f)(n c (id c )) holds for all f Hom(c, c ) = h c (c ). Thus the natural transformation is determined by N c (id c ) only, which proves the injectivity of the Yoneda map. Concersely, any element x F (c) yields a map of sets Hom C (c, c ) = h c (c ) F (c ) f x N(f) := F (f)(x) which compose to a natural transformation x N : y c F of functors C Set. This can be seen by checking for any morphism c f c that the diagram Hom(c, c ) x N c F (c ) f F (f) Hom(c, c ) x N c F (c ) commutes, since the two relevant morphisms applied to ϕ Hom(c, c ) yield F (f ϕ)(x) and F (f)f (ϕ)(x) respectively. Corollary

18 1. In the special case F = h c = Hom C (c, ), we have Nat(h c, h c ) = h c (c) = Hom C (c, c). In particular, the functors h c = Hom C (c, ) and h c = Hom C (c, ) are isomorphic, if and only if the objects c and c are isomorphic. If a functor F : C Set is representable, i.e. isomorphic to a functor h a = Hom C (a, ), then the representing object a C is determined up to isomorphism. 2. Let X be a simplicial set and Δ n = Hom Δ (, [n]) be the standard simplex from example Then X n = Hom sset (Δ n, X ). Remarks The first statement shows that the functor h : C opp Fun(C, Set) and similarly h : C Fun(C opp, Set) is fully faithful. It is called the Yoneda embedding and allows to identify C with a subcategory of presheaves on C. 2. This justifies the name standard n-simplex for the simplicial set Δ n in example An element x X n was called an n-simplex of the simplicial set X. It can now be characterized as a map Δ n X of simplicial sets, which stresses the analogy to the singular simplicial set associated to a topological space. Proof. The first assertion is evident. For the second assertion, consider X Fun(Δ opp, Set) and recall that Δ n = Hom(, [n]) as a functor from the category of finite ordinals to Set. Then apply the Yoneda lemma Hom sset (Δ n, X ) def = Nat(Hom(, [n]), X) Yoneda = X([n]) = X n. Observation We draw one important consequence from the existence of a product on the category sset of simplicial sets. Recall that the Cartesian product gives a product on the category Set of sets that obeys the adjunction relation Hom Set (A B, C) = Hom Set (A, Hom Set (B, C)). This makes sense, because Hom Set (B, C) is in the category we consider. 2. Given two simplicial sets X and Y, we would like to find an internal Hom, i.e. a simplicial set Hom(X, Y ) such that Hom sset (Z, Hom(X, Y )) = Hom sset (Z X, Y ). 3. Applying this specifically to the standard n-simplex Z := Δ n, we find Hom(X, Y ) n = Hom(Δ n, Hom(X, Y )) = Hom sset (Δ n X, Y ) 16

19 that the set of standard simplices of Hom(X, Y ) has to be Hom sset (Δ n X, Y ). In particular, we find We define the face maps Hom(X, Y ) 0 = Hom sset (Δ 0 X, Y ) = Hom sset (X, Y ). d i : Hom(X, Y ) n = Hom sset (Δ n X, Y ) Hom(X, Y ) n 1 = Hom sset (Δ n 1 X, Y ) as mapping f Hom sset (Δ n X, Y ) to Δ n 1 X di id X Δ n X f Y The degeneracy maps are defined analogously. Proposition For any three simplicial sets, the adjunction relations and hold. They are natural in all arguments. 1.2 The Kan Property Hom sset (Z, Hom(X, Y )) = Hom sset (Z X, Y ). Hom(Z, Hom(X, Y )) = Hom(Z X, Y ) We introduce two subcomplexes of the standard simplex Δ n = Hom Δ (, [n]). Recall that in degree k, the set Δ n k consists of all non-decreasing sequences of length k with values in [n], i.e. {(x 0,..., x k ) [n] k+1 0 x 0... x k n}. We denote by ι n = (0, 1,..., n) Δ n n the single non-degenerate n-simplex on Δ n. Definition The simplicial set Δ n for n 0 is the smallest subcomplex of Δ n containing the faces d j (ι n ) with j = 0,..., n of the distinguished n-simplex. We set Δ 0 =, the simiplicial set with in each degree. (This object is initial for sset, i.e. for any simplicial set X, there is a single morphism X.) Remarks Explicitly, one finds that for 0 j n 1, for the set of j-simplices on the simplicial set Δ n the set ( Δ n ) j = Δ n j. This holds, since Δ n n 1 contains d i (0,..., n) = (0, 1,..., î,... n) for any i. (Here, a hat indicates that this index has to be dropped.) Applying iteratively face and degeneracy operators shows the claim. For j n, the set Δ n j only contains iterated degeneracies of elements of Δ n k with 0 k n 1. Put differently, the simplicial set Δ n contains all simplices x : [k] [n] for which the map is not surjective. 2. We can write Δ as a coequalizer of simplicial sets: n Δ n 2 Δ n 1 ( Δ n ). 0i<jn i=0 where the two arrows implement the simplicial relations d j d i = d i d j 1 for i < j. Informally, we take (n+1)-many (n 1) standard simplices for the faces and then identify them along (n 2)-simplices. Explicitly, if the (n 1)-simplices are indexed by 0 i n, i.e. Δ n 1 i and the (n 2)-simplices Δ n 2 ij by pairs i < j, then the two maps are composed of Δ n 2 ij d i Δ n 1 j and Δ n 2 d j 1 ij Δ n 1 i respectively, with the maps d k given as in (2). 17

20 Definition The k-th horn Λ n k for n 0 is the subcomplex of Δn and thus of Δ n that is generated by all faces of Δ n, except for the face (0, 1,..., k,... n) opposite the k-th vertex. Remarks Explicitly, the simplicial set Λ n k contains all simplices x : [l] [n] for which Im(x) does not contain the set {0, 1,..., k,..., n}. The k-th horn Λ n k is thus obtained from Δn by removing the k-th face d k (ι n ), i.e., the face opposite to vertex k. 2. More formally, the horn Λ n k is defined as the following coequalizer Δ n 2 Δ n 1 Λ n k. i k 0i<jn For details, see [Goerss-Jardine, Lemma I.3.1]. Recall from remark that an n-simplex x X n of a a simplicial set X could be identified by the Yoneda lemma with a morphism Δ n X of simplicial sets. In the same way, we can now define cycles and horns of simplicial sets. Definition Let X be a simplicial set. 1. An n-cycle on X is a morphism Δ n+1 X of simplicial sets. 2. A Λ n k -horn on X is a morphism Λ n k X of simplicial sets. Remark It is enough to specify a simplicial map f : X Y on non-degenerate simplices. Then the condition f(s i x) = s i f(x) determines the map on degenerate simplices. This allows us to unpack the definitions. 1. To specify an n-cycle on a simplicial set X, we have to specify the images of (n+2)-many n-simplices, i.e. to determine an (n + 2) tuple (x 0,..., x n+1 ) (X n ) n+2 of n-simplices of X. Since a boundary is a coequalizer, we have to require that it satisfies d i x j = d j 1 x i for all 0 i < j n + 1. (Note that here the index n refers to the highest degree in which a non-degenerate simplex appears.) 2. An n-cycle on X is called an n-boundary, if there is an (n + 1)-simplex x X n+1 such d i x = x i for all i = 0,... n + 1. The simplicial relation d i d j x = d j 1 d i x for i < j implies that every boundary is a cycle. We say that an n-boundary is filled by an (n + 1)-simplex. 3. An n-tuple (x 0,..., x k 1,?, x k+1,..., x n ) of (n 1)-simplices, where the kth simplex is not specified yet, is an (n, k)-horn on X, if it is compatible in the sense of 1. We now need notions of completing and filling horns. Definition Let X be a simplicial set and Λ n k X a horn on X. 18

21 1. A cycle Δ n+1 X on X is called a completion of the horn, if the diagram commutes. Λ n k X Δ n 2. An n-simplex x X n+1 is called a filling of the horn Λ n k, if the diagram commutes. Λ n k X Δ n+1 Remark A completion of the horn (x 0,..., x k 1,?, x k+1,..., x n+1 ) on X is an n-simplex x k on X such that the (n + 2)-tuple (x 0,..., x n+1 ) is a cycle in the sense of remark A filling of the horn (x 0,..., x k 1,?, x k+1,..., x n+1 ) is an (n+1)-simplex x whose boundary (d 0 x,..., d n+1 x) is a completion of the horn, i.e. d i x = x i for i = 0, 1,..., ˆk,... n + 1. We now discuss the situation more specifically for simplicial sets that are nerves of categories: Observation Let C be a category and N(C) its nerve. 1. In dimension n = 2, horns λ: Λ 2 k respectively look like N(C) on the simplicial set N(C) for 0 k 2 hence c 0 c 1 c 1 f f g c 2, c 0 c 2, c 0 h h c 1 g 2. Using the composition h = g f we can uniquely extend any horn λ: Λ 2 1 N(C) on N(C) to an entire 2-simplex σ : Δ 2 N(C), i.e., given (f, g), there is a unique dashed arrow making the diagram of simplicial sets c 2. Λ 2 1 Δ 2 (f,g)!σ N(C) commute. The composition is given by the new face d 1 (σ): Δ 1 N(C). 3. If instead we consider a horn λ: Λ 2 0 N(C) in the special case that h = id is an identity morphism, then the existence of an extension to a 2-simplex is equivalent to the existence of a left inverse to f. Similar observations can be made for horns λ: Λ 2 2 N(C), leading to a right inverse. 19

22 4. This different behavior extends to higher dimensions and motivates the following terminology: the horns Λ n k, 0 < k < n, are inner horns while the extremal cases Λn 0 and Λ n n are outer horns. It turns out that these horn extension properties are suitable to describe the essential image of the nerve functor. We denote by Grpd the category of groupoids, cf. example Theorem Let K be a simplicial set. 1. There is an isomorphism K = N(C) of simplicial sets for some C Cat, if and only if every inner horn Λ n k K, 0 < k < n, on K can be uniquely extended to an n-simplex Δ n K on K. 2. There is an isomorphism K = N(Γ) for some Γ Grpd, if and only if every horn Λ n k K, 0 k n with n 2 can be uniquely extended to an n-simplex Δ n K on K. Proof. We first assume that K is the nerve of a small category C. Let f 0 : Λ n i X for 0 < i < n be a map of simplicial sets. We have to show that f 0 can be uniquely extended to a map f : Δ n K. Let X k C for 0 k n be the image of the vertex (k) Λ n i and g k : K k 1 K k be the morphism in C determined by the restriction of f 0 to (k 1, k). The composable chain of morphisms X 0 g 1 X1 g 2... g n Xn determines an n-simplex f : Δ n K on K. If this is a solution to the horn extension problem, it is unique, since any other extension restricting to the same map on the horn gives rise to the same chain of morphisms. It can be checked directly that this solves the extension problem. We now prove the converse and assume that the simplicial set K obeys the unique horn extension property for inner horns. We have to show that then K is isomorphic to the nerve of a small category C which we construct as follows: (i) The objects of C are the vertices of K, i.e. elements of K 0 or, equivalently, maps Δ 0 K. (ii) Given a pair of objects x, y C, we let Hom C (x, y) denote the collection of all edges e : Δ 1 K such that e (0) = x and e (1) = y. (iii) Let x be an object of C. Then the identity morphism id x is the edge of K defined by the composition Δ 1 Δ 0 x K. (iv) Let f : x y and g : y z be morphisms in C. Then f and g together determine a map σ 0 : Λ 2 1 K. This is an inner horn, and thus the map can be extended uniquely to a 2-simplex σ : Δ 2 K on K. We define the composition g f to be the morphism from x to z in C corresponding to the edge given by the composition We have to verify the axioms of a category: Δ 1 d 1 Δ 2 σ K. 20

23 (a) For every object y C, the identity id y is a unit with respect to composition. In other words, for every morphism f : x y in C and for every morphism g : y z in C, we have id y f = f and g id y = g. These equations are consequences of the degenerate 2-simplices s 1 (f), s 0 (g) Hom sset (Δ 2, K ). (b) Composition is associative. That is, for every sequence of composable morphisms w f x g y h z we have h (g f) = (h g) f. To prove this, introduce the 2-simplices σ 012 := x σ 123 := f g g f w y x y g h h g z σ 023 := y g f h h (g f) w z which together define a map τ 0 : Λ 3 2 K. Since this is an inner horn, we can extend uniquely to a 3-simplex τ : Δ 3 K. Its forth face is which exactly shows associativity. x f h g h (g f) w z We thus have a well-defined category C. We also have a map φ : K N(C) of simplicial sets: it is clear on 0 and K 1 and we define it inductively since we know it on horns and can extend it using the unique horn filling condition. We show by induction that that φ is an isomorphism: φ n : Hom sset (Δ n, K) Hom sset (Δ n, N(C)) is a bijection. For n = 0, 1, this follows from th cinstruction. Assume thus that n 2 and chose an integer 0 < i < n. Consider the commuting diagram Hom sset (Δ n, K) Hom sset (Δ n, N(C)) Hom sset (Λ n i, K) Hom sset (Λ n i, N(C)) Since K and the nerve N(C) both satisfy the unique extension condition (for the nerve, this was shown in the first part of the proof) the vertical maps are bijective. It will therefore suffice to show that the lower horizontal map is bijective, which follows from the induction hypothesis, since only non-degenerate simplices of lower degree are involved. The statement about groupoids follows as sketched in Observation The characterization of the essential image of the nerve functor N : Grpd sset can weakened concerning the uniqueness of the fillers. It then yields an important notion: Definition A simplicial set X is a Kan complex if every horn Λ n k X on X for 0 k n can be extended to an n-simplex Δ n+1 X on X. (This is called the horn filling condition.) 21

24 Remarks We reformulate the Kan condition: The simplicial set X satisfies the Kan condition if for any collection of (n 1)-simplices x 0,..., x k 1, x k+1,..., x n in X such that d i x j = d j 1 x i for any i < j with i k and j k, there is an n-simplex x in X such that d i x = x i for all i k. 2. We will see that the singular simplicial set of a topological space obeys the Kan condition. 3. The standard simplices Δ n, n > 0, do not satisfy the Kan condition. Consider the horn Λ 2 0 Δ 1 on Δ 1 that is defined by mapping the non-degenerate 1-simplex (0, 2) of Λ 2 to the degenerate simplex (0, 0) of Δ 1 and the non-degenerate 1-simplex (0, 1) of Λ 2 0 to the simplex (0, 1) of Δ 1. There is a unique simplicial map Λ 2 0 Δ 1 with these properties. This cannot be extended to a map Δ 2 Δ 1, since for this map we would need 0 0, 1 1, and 2 0, which is clearly not order-preserving on Δ For the same reason, no ordered simplicial complex X (augmented to be a simplicial set) can satisfy the Kan condition, unless K is a discrete set of points. 5. In particular, the simplicial sets Δ n, Δ n and Λ n k the trivial complex does. do not satisfy the Kan property, whereas 6. A quotient of two Kan complexes does not have to be a Kan complex again. The following lemma is a first indication that Kan complexes form a good subcategory: Proposition A simplicial group G, i.e. a simplicial object in the category of groups, is a Kan complex when considered as a simplicial set. A fortiori, simplicial abelian groups and simplicial modules over a ring are Kan complexes, when considered as a simplicial set. Proof. Suppose we are given a horn (x 0,..., x k,..., x n+1 ) with x i G n such that d i x j = d j 1 x i for i < j and need to find g G n+1 such that d i g = x i. We use induction to find g r G n+1 such that d i g r = x i for all i r with i k and start our induction with g 1 = e. Suppose, g r 1 has been found. If r = k, then g k = g k 1, i.e. do not change anything. In all other cases, consider u := x 1 r (d r g r 1 ) G n. For i < r and i k, we find d i u = (d i x r ) 1 (d i d r g r 1 ) = (d i x r ) 1 (d r 1 d i g r 1 ) = (d i x r ) 1 (d r 1 x i ) = e where in the last step, we used the defining equations of a horn. Thus d i s r u = s r 1 d i u = s r 1 e = e for i < r. We set g r := g r 1 (s r u) 1. This element has to obey d i g r = x i for all i r. Indeed, we find d r g r = d r g r 1 (d r s r u) 1 = (d r g r 1 )u 1 = x r and for i < r d i g r = (d i g r )(d i s r u)) 1 = d i g r = x i. Lemma Arbitrary (co-)products of Kan complexes are again Kan complexes. 22

25 Proof. We restrict our attention to the case of a product of two complexes. The case of an arbitrary index set is very similar and the coproduct case is left as an exercise. Let (z 0,...,?,..., z n ) be a horn in (K L). The definition of the product implies that z i = (x i, y i ); by the definition of the face operators in a product, (x 0,...,?,..., x n ) and (y 0,...,?,..., y n ) are compatible in K and L, respectively. Using the Kan properties we can complete both horns separately by some x k and y k ; then z k := (x k, y k ) completes the first horn. Denoting by Kan sset the full subcategory spanned by the Kan complexes, we thus have the following commutative diagram of fully faithful functors Grpd N Kan Cat N sset. As a summary, Kan complexes and nerves of categories are simplicial sets with certain horn extension properties, but these properties differ in two important aspects. 1. First, in a Kan complex all horns can be extended, while in the nerve of a category this is, in general, only the case for inner horns. 2. Second, for Kan complexes we have a mere existence statement while for nerves of categories the extensions are unique. This uniqueness property will be dropped for - categories: it suffices that compositions exist and that the actual choice of compositions is homotopically irrelevant. This is similar to the concatenation of paths in a topological space: there is no preferred composition law for paths parametrized by the unit interval that would be associative and unital. But paths can be glued and the actual choice of the parametrization is homotopically irrelevant, e.g., all choices are homotopic. Definition A simplicial set C is an -category, if every inner horn Λ n k C, 0 < k < n, can be extended to an n-simplex Δ n C. 2. A simplicial set C is an -groupoid, if every horn Λ n k C, 0 k n, can be extended to an n-simplex Δ n C. Thus spaces and ordinary categories give rise to -categories. The following interesting example of an -category is not of this form: Remark The prototypical example of -category arises from homotopy theory. For any topological space X, there is an -category π (X), with objects given by the points of X, 1-morphisms given by continuous paths in X, 2-morphisms given by homotopies of paths with fixed end-points, 3-morphisms given by homotopies between homotopies, and so on. Since the composition of paths is only associative up to homotopy, i.e. up to a 2-morphism, π (X) is necessarily a weak -category. Nevertheless, the 2-morphism above, which is part of the data, is invertible up to 3-morphisms. Indeed, all k-morphisms in π (X) are invertible, hence it is an -groupoid. 23

26 2 Connections to Topology 2.1 Geometric realization We now relate more closely topological spaces and simplicial sets and construct a functor : sset Top. This functor should associate to a standard simplex Δ n sset the topological standard simplex Δ n, justifying our notation. Definition For the topological standard n-simplex Δ n with n N 0 Δ n := {(v 0,..., v n ) R n+1 v i 0 and n i=0 v i = 1}. we consider the maps δ i : Δ n 1 Δ n and σ i : Δ n+1 Δ n for i [n] given by δ i (v 0,..., v n 1 ) := (v 0,..., v i 1, 0, v i,..., v n 1 ) and σ i (v 0,..., v n+1 ) := (v 0,..., v i + v i+1,..., v n+1 ). The map δ i embeds a topological (n 1) simplex as the i-th face of a topological n-simplex. The maps σ i collapse topological simplices. We note that the standard simplices Δ n = Hom Δ (, [n]) form a full subcategory of sset that is, by the Yoneda lemma, isomorphic to the category of finite ordinals Δ. Conversely, sset = Fun(Δ, Set) can be seen as a completion of the category Δ. The following proposition is now obvious. Proposition The maps Δ n Δ n and d i δ i, s i σ i define a functor : Δ Top. For a general simplicial set, we throw in for each element x X n a topological simplex Δ n as suggested by called x X n an n-simplex. More formally, we endow X n with the discrete topology and consider the disjoint union n N 0 X n Δ n. Now we have to identify the faces of the topological simplices as dictated by the simplicial set X. This leads to the following definition: Definition The geometric realization X of a simplicial set X is the topological space given as the quotient space ( ) / X n Δ n n N 0 where the equivalence relation is generated by (d i x, v) (x, δ i v) and (s i x, w) (x, σ i w) for all n N, x K n, v Δ n 1, w Δ n+1, i [n]. It is clear that this extends to a functor sset Top. We have thus solved the problem of extending the functor Δ Top sset 24

27 Examples Consider the simplicial set Δ 0 = Hom Δ (, [0]). Since the singleton set [0] is terminal, it has a single n-simplex in each degree. We thus start with a collection of topological simplices, one of each dimension. We build up using only face maps to get a point, a circle, a dunce cap [Bredon, p. 50] which is obtained by identifying all faces and all vertices of a triangle and so on. The degeneracy maps, however, collapse this to a single point. This example shows that it is important to include degeneracy maps. 2. The geometric realization of the simplicial set Δ n is homeomorphic to a sphere. Indeed, we glue together (n + 1)-many non-degenerate (n 1)-simplices along (n 2)-simplices 3. This is not the only way to obtain the sphere S n 1 for n 2 as the geometric realization of a simplicial set. Let X be a simplicial set whose only nondegenerate simplices are denoted by [0] X 0 and [0,..., n 1] X n 1. All simplices in X i for 0 < i < n 1 are the degenerate simplices [0,..., 0]. This, of course, forces all of the faces of the nondegenerate simplex [0,..., n 1] to be [0,..., 0], and we see that the realization X is equivalent to the standard construction of S n 1 as a CW complex by collapsing the boundary of an (n 1)-cell to a point. 4. In the special case of the category BG, one obtains the classifying space of a group. The extension can be characterized as a Kan extension. Definition Δ Top sset 1. Given functors F : C E and K : C D, a left Kan extension of F along K is a functor Lan K F : D E, together with a natural transformation η : F Lan K F K, as a diagram C K F η D E Lan K F such that any other such pair (G : D E, γ : F G K) factors uniquely through η. As diagrams: F C E K F C E γ = η Lan K F G K α D D G 2. Dually, a right Kan extension is a functor Ran K F : C E, together with a natural transformation ɛ : Ran K F K F, as a diagram F C ɛ K D 25 E Ran K F

28 such that any other such pair (G : D E, γ : G K F ) factors uniquely through ɛ. As diagrams: F C E K F C E γ = ɛ Ran K F G K α D D G Remark Kan extensions will not always exist. Definition Let Γ be an (essentially) small category and C be a category. 1. A functor F : Γ C is called a diagram of shape Γ with values in C. 2. A cone to F is an object N C, together with a family (ψ γ : N F (γ)) γ Γ of morphisms indexed by the objects γ Γ, such that for every morphism f : γ γ in Γ, we have F (f) ψ γ = ψ γ. 3. A limit of the diagram F : Γ C is a cone (L, φ) to a diagram F such that for any other cone (N, ψ) to F there exists a unique morphism u : N L such that φ γ u = ψ γ for all γ Γ. 4. The definition of a colimit is dual. Example Let D = 1 be the category with one object and one morphism, the identity on this object. This category is terminal in the sense that for any category C, there is only one functor K : C 1. A functor 1 E is given by an object e E. The composition C K 1 Lan K E is thus a constant functor Δ e : C E, sending any c C to a fixed object e E and any morphism to the identity id e on e. Given a left Kan extension of F : C E along a functor K : C 1, we have a natural transformation η : F Δ e. Such a natural transformation is called a cone under F with vertex e. A natural transformation between two functors e, e : 1 E is a morphism e e in E. Thus in this case the left Kan extension is the colimit of the functor F. Dually, a right Kan extension along a functor K : C 1 is a limit. Remarks Let Γ be a discrete category, i.e. a category with only identity arrows. A diagram of shape Γ is then a family of objects (V γ ) γ Γ. A cone is then a family of maps (f γ : Y X γ ) γ Γ. The universal property of the product states a bijection F Hom C (Y, γ X γ ) γ Hom C (Y, X γ ) by postcomposing with the canonical projections. Thus, the product is a limit; similarly, the colimit is the coproduct of this family. 26

29 2. In the definition of a pullback (and of a pushout), Γ was a category with 3 objects of a hook form. For example, the partially ordered set Ω = {0, 1, 2} with 0 < 1, 0 < 2, but not 1 2 gives a category with non-identity arrows γ 0 γ 1 and the colimits are pushouts. Similarly, the pullback is a limit. γ 2 3. The direct or inductive limit of topological spaces is a colimit; the projective or inverse limit of topological spaces is a limit, cf. [Top-WS15, Section 1.14]. Proposition Let C be a category and X : Γ C a diagram of the form Γ. If in C difference cokernels and coproducts exist, then X has a colimit which can be expressed as a difference kernel γ 1 γ 2 X(γ 1 ) f g γ X(γ) The structure maps are compositions of injections into the coproduct, followed by the map to the cokernel. A dual statement holds for limits. Definition A category is called (finitely) cocomplete, if every (essentially finite) diagram in C has a colimit. 2. A category C is called (finitely) complete, if every (essentially finite) diagram in C has a limit. 3. A functor F : C D is called cocontinuous (or preserves colimits), if for any universal cocone X Δ T ) in C the induced cocone F (X) F (Δ T ) is universal as well. 4. The notion of a continuous functor is dual. Remarks A cocontinous functor can be characterized equivalently by the conditon that for any diagram X in C with a colimit colim(x), also the diagram F (X) in D has a colimit and the canonical morphism colim(f (X)) F (colim(x)) is an isomorphism. 2. The following categories are complete and cocomplete: Set, Ab, Grp, R-mod and Top, since they have difference (co-)kernels and (co-)products. The categories of finite sets, finite abelian groups and finite-dimensional vector spaces are finitely complete and finitely cocomplete, but neither complete nor cocomplete. 27

30 3. The forgetful functor U : Ab Set is not cocontinous since the map U(A 1 ) U(A 2 ) U(A 1 A 2 ) = U(A 1 A 2 ) = U(A 1 ) U(A 2 ) is not surjective. (Indeed, the initial object, the colimit of the empty diagram, is not preserved.) Forgetful functors can not be expected to be cocontinuous in general. (The forgetful functor Top Set is continuous, though.) 4. If Γ is a small category, then the functor category ˆΓ := Fun(Γ opp, Set) is cocomplete. The Yoneda embedding Γ ˆΓ induces [Brandenburg, Proposition ] for any cocomplete category D an equivalence of categories Fun c (ˆΓ, D) = Fun(Γ, D). Here Fun c denotes the full subcategory of cocontinuous functors. 5. Any representable functor C opp Set is continuous. For any diagram X in C with colimit colimx in C, the canonical map Hom(colimX, Y ) lim Hom(X, Y ) is an isomorphism for any Y C. (Note that a colimit in C is a limit in C opp.) Similarly, the canonical map Hom C (Y, lim X) lim Hom(A, X) is an isomorphism for any A C. Indeed, the limit in the category Set is a subset of the product i I X(i) consisting of tuples (x i ) such that X(i j)x i = x j. Now use the universal property of the product. Definition Consider two (essentially) small categories Γ 1 and Γ 2. Their tensor product is the category Γ 1 Γ 2 with objects Obj(Γ 1 ) Obj(Γ 2 ) and morphisms freely generated from (γ 1, γ 2 ) (γ 1, γ 2) for a morphism γ 2 γ 2 and any γ 1 Γ 1, and (γ 1, γ 2 ) (γ 1, γ 2 ) for a morphism γ 1 γ 1 and any γ 2 Γ A diagram X of the form Γ 1 Γ 2 is called admissible, if for any pair of morphisms γ 1 γ 1 and γ 1 γ 2 the induced diagram X(γ 1, γ 2 ) X(γ 1, γ 2) commutes. X(γ 1, γ 2 ) X(γ 1, γ 2) Note that this implies that any morphism γ 1 γ 1 induces a morphism of diagrams X(γ 1, ) X(γ 1, ) of shape Γ 2. Consider an admissible diagram X of shape Γ 1 Γ 2. Suppose, the limit of the diagram X(γ 1, ) of shape Γ 2 exists for all γ 1 Γ 1. We denote it by lim γ2 Γ 2 X(γ 1, γ 2 ). Any morphism γ 1 γ 1 now induces a morphism of diagrams and, by the universal property, a morphism of limits lim X(γ 1, γ 2 ) lim X(γ γ 2 Γ 2 γ 2 Γ 2 1, γ 2 ). We thus obtain a diagram lim γ2 Γ 2 X(, γ 2 ) of shape Γ 1. 28

31 Proposition The cones for the diagram lim γ2 Γ 2 X(, γ 2 ) of shape Γ 1 are in bijection to cones for the diagram X of shape Γ 1 Γ 2. In particular, lim γ1 Γ 1 lim γ2 Γ 2 X(γ 1, γ 2 ) exists, if and only if lim (γ1,γ 2 ) Γ 1 Γ 2 X(γ 1, γ 2 ) exists and the two limits are isomorphic, lim γ 1 Γ 1 lim X(γ 1, γ 2 ) = lim X(γ 1, γ 2 ). γ 2 Γ 2 (γ 1,γ 2 ) Γ 1 Γ 2 Proof. A cone to the diagram lim γ2 Γ 2 X(, γ 2 ) of shape Γ 1 consists of morphisms such that the diagramm T lim γ2 Γ 2 X(γ 1, γ 2 ) lim γ2 Γ 2 X(γ 1, γ 2 ) commutes for all γ 1 γ 1. For fixed γ 1, a morphism T lim γ2 Γ 2 X(γ 1, γ 2 ) is a family of morphisms T X(γ 1, γ 2 ) for γ 2 Γ 2 such that for all γ 2 γ 2 the diagram T X(γ 1, γ 2 ) X(γ 1, γ 2) commutes. Combining the two diagrams, we get a cone on the diagram of shape Γ 1 Γ 2. Since the construction is natural on T, the universal cones are in bijection as well. Corollary (Limits commute). Given an admissible diagram X of shape Γ 1 Γ 2, we have lim γ 1 Γ 1 lim X(γ 1, γ 2 ) = lim γ 2 Γ 2 γ 2 Γ 2 lim X(γ 1, γ 2 ), γ 1 Γ 1 provided the limits exist. In particular, products commute. A dual statement holds for colimits. Proof. From the preceding proposition, we conclude that both sides are limits of the diagram X of shape Γ 1 Γ 2. Kan extensions are closely linked to adjoint functors: Definition Let C and D be any categories. A functor F : C D is called left adjoint to a functor G : D C, if for any two objects c C and d D, there is an isomorphism of Hom-spaces with the following natural property: Φ c,d : Hom C (c, Gd) Hom D (F c, d) 29

32 For any homomorphism c f g c in C and d d in D consider for ϕ Hom D (F c, d) the morphism Hom(F f, g)(ϕ) := F c F f F c ϕ d g d Hom D (F c, d ) and for ψ Hom C (c, Gd) the morphism Hom(f, Gg)(ψ) := c f c ψ Gd Gg Gd Hom C (c, Gd ). The naturality requirement is then the requirement that the diagram Hom C (c, Gd) Φ c,d Hom D (F c, d) Hom(f,Gg) Hom C (c, Gd ) Φ c,d Hom(F f,g) HomD (F c, d ) commutes for all morphisms f, g. 2. We write F G and also say that the functor G is a right adjoint to F. Examples The forgetful functor U : vect(k) Set, which assigns to any K-vector space the underlying set has as a left adjoint, the freely generated vector space on a set: F : Set vect(k), Indeed, we have for any set M and any K-vector space V an isomorphism Φ M,V : Hom Set (M, U(V )) Hom K (F (M), V ) ϕ Φ M,V (ϕ) where Φ M,V (ϕ) is the K-linear map defined by prescribing values in V on the basis of F (M) using ϕ and extending linearly: Φ M,V (ϕ)( m M λ m m) := m M λ m ϕ(m). In particular, we find the isomorphism of sets Hom Set (, U(V )) = Hom K (F ( ), V ) for all K-vector spaces V. Thus Hom K (F ( ), V ) has exactly one element for any vector space V. This shows F ( ) = {0}, i.e. the vector space freely generated by the empty set is the zero-dimensional vector space. 2. In general, freely generated objects are obtained as images under left adjoints of forgetful functors. It is, however, not true that any forgetful functor has a left adjoint. As a counterexample, take the forgetful functor U from fields to sets. Suppose a left adjoint exists and study the image K of the empty set under it. Then K is a field such that for any other field L, we have a bijection Hom F ield (K, L) = Hom Set (, U(L)) =. Since morphisms of fields are injective, such a field K would be a subfield of any field L. Such a field does not exist. 30

33 Observation Let F G be adjoint functors. From the definition, we get isomorphisms and Hom C (G(d), G(d)) = Hom D (F (G(d)), d) Hom D (F (c), F (c)) = Hom C (c, G(F (c))). The images of the identity on G(d) and F (c) respectively form together natural transformations ɛ : F G id D and η : id C G F. Note the different order of the functors F, G in the composition and compare to the definition of a duality. These natural transformations have the property that for all objects c in C and d in D the morphisms G(d) η G(d) (GF ) G(d) = G(F G)(d) G(ɛ d) G(d) and are identities. F (c) F (η c) F (GF )(c) = (F G)F (c) ɛ F (c) F (c) 2. Conversely, we can recover the adjunction isomorphisms Φ c,d from the natural transformations ɛ and η by and their inverses by Hom C (c, G(d)) F Hom D (F (c), F (G(d)) (ɛ d) HomD (F (c), d) Hom D (F (c), d) G Hom C (G(F (c)), G(d)) η d Hom C (c, G(d)). 3. Note that a pair of adjoint functors F G is an equivalence of categories, if and only if ɛ and η are natural isomorphisms of functors. Proposition Any left adjoint functor is cocontinous. Any right adjoint functor is continuous. Proof. Given an adjunction F G, with F : C D and G : D C, we have a natural isomorphism Φ : Hom D (F, ) Hom C (, G ). Let X be a diagram of shape Γ in C with universal cocone (X γ T ). Now let (F (X γ ) Y ) be any cocone of the diagram F X of shape Γ in D. Since Φ is natural, we obtain a cocone (X γ G(Y )) over the original diagram in C. The universal property of the universal cocone gives us a unique morphism T G(Y ) such that X γ T G(Y ) equals X i G(Y ) for all γ Γ. Via Φ 1, this gives a unique morphism F (T ) Y such that F (X γ ) F (T ) Y equals F (X γ ) Y. Thus F (X γ ) F (T ) is a universal cocone and F is cocontinuous by remark Remark

34 Assume, a right Kan extension exists for all functors F : C E. As a consequence of the universal property, any natural transformation α : G F factors uniquely to a natural transformation α : Ran K G Ran K F such that α ɛ G = α ɛ F : We write Ran K (α) := α. F G C E K ɛ G Ran K G D = F C ɛ K F D E Ran K F α Ran K G Given functors F, G, H : C E and natural transformations H β G α F the universal property implies that Ran K (β α) = Ran K (β) Ran K (α). We thus obtain a functor Ran K : Fun(C, E) Fun(D, E). Similarly, if all functors C D have a left Kan extension, we have a functor We now relate Kan extensions to adjunctions: Lan K : Fun(C, E) Fun(D, E). Theorem The functors Lan K and Ran K are respectively the left and right adjoints to the functor K : Fun(D, E) Fun(C, E) defined on the functor categories via precomposition with K : C D. Proof. Consider two functors Nat D,E (Lan K F, ), Nat C,E (F, K) : Fun(D, E) Set. The universal property of the left Kan extension (Lan K F, η) says that the natural transformation η : Nat D,E (Lan K F, ) Nat C,E (F, K) is a natural isomorphism. Thus, the functor Lan K F represents the functor Nat C,E (F, K) : Fun D,E Set. Similarly, Ran K F represents Nat C,E ( K, F ). But the first statement means that, for any F : C E, we have Fun D,E (Lan K F, G) = Fun C,E (F, G K) 32

35 v vectfd Endk (v) = k natural in G. Thus Lan K K. The morphisms η define the components for the unit η : id ( K) Lan K of the first adjunction. Indeed, the right hand side is the functor Fun C,E Lan K FunD,E K Fun C,E G (Lan K G) K Similar arguments show that the right Kan extension provides a right adjoint, K Ran K. Example Since adjoints are unique up to unique isomorphism, the left adjoint to any precomposition functor K, if it exists, will be a left Kan extension functor along K. Similarly the right adjoint to any precomposition functor will be a right Kan extension functor. 2. A special case is given by taking a group homomorphism ϕ : H G which gives a functor ι : //H //G of groupoids and by precomposition a functor Fun( //G, vect) = rep(g) ι Fun( //H, vect) = rep(h) which is just restriction. Then the left adjoint is induction and the right adjoint coinduction. For a proof and more about Kan extensions, we refer e.g. to [Lehner]. An elegant formula for a left Kan extension can be given in terms of coends: Definition Given a functor F : A A op B, a dinatural transformation from F to an object B B is a family of morphisms ϕ X : F (X, X) B for X A such that ϕ Y F (id Y, g) = ϕ X F (g, id X ) for all g Hom A (X, Y ). 2. A coend (C, ι) for F is an object C B with a dinatural transformation ι that is universal among all dinatural transformations from F to an object of B in the sense that for any such family ϕ there exists a unique morphism κ obeying ϕ X = κ ι X for all X A. Both the pair (C, ι) and the object C are denoted by X A F (X, X). 3. An end is defined dually. Remarks Let vect fd be the category of finite-dimensioanl vector spaces over a given field k. A useful exercise is to show that the coend of the functor Hom : vect fd vect opp fd vect fd W V Hom k (V, W ) with structure morphisms End k (v) k given by the trace. 33

36 2. If D is a cocomplete category, then the coend of a functor F : A A opp D exists and can be expressed as the difference cokernel α:a a F (a, a) f g a F (a, a). Indeed, use the universal property of the coproduct to find morphisms δ, δ such that for all α : a a the diagrams α:a a F (a, a) δ a F (a, a) F (a, a ) F (id a,α) F (a, a) and α:a a F (a, a) δ a F (a, a) commute. F (a, a ) F (α,id a ) F (a, a ) 3. Dually, an end can be expressed as an equalizer. This entails the important formula for the natural transformations between two functors F, G : C D: Hom(F (c), G(c)) = Nat(F, G). c C Indeed, the components of a natural transformation α : F G are an element (α c ) c C c Hom D(F (c), G(c)) in the product, and the structure map of the end is α (α c ). Observation We assume that in a category B arbitrary coproducts exist. Given a set M and an object X B, we let M X := m M X B. 2. Any object X B thus gives a functor X : Set B. This functor is left adjoint to the functor Hom B (X, ) : B Set. Indeed, we find for any set M and any pair of objects X, Y B Hom B (M X, Y ) def = Hom B ( m M X, Y ) = m M Hom B(X, Y ) = Hom Set (M, Hom B (X, Y ))). Here we used the universal property of the coproduct and the fact that for any set Hom Set (M, X) = m M X. 34

37 3. We conclude that the functor X : Set B is cocontinuous, since it is a left adjoint. We have obtained a functor B Fun c (Set, B) X X where Fun c denotes the cocontinuous functors. Conversely, given a cocontinuous functor ϕ : Set B, its image ϕ( ) B on the singleton set gives an object in B. Obviously, X = X and for F : Set B cocontinuous we have for all M Set F (M) = F ( m M ) = m M F ( ) = M F ( ) so that F = F ( ). Thus the two functors are quasi inverses and we conclude Fun c (Set, B) = B. Proposition For any functor F : A C into a cocomplete category, there is a natural isomorphism a A Hom A (a, ) F (a) = F. Proof. Indeed, for any x A and c C, we have Hom C ( a A Hom A (a, x) F (a), c) = Hom C(Hom a A A (a, x) F (a), c) = = a A Hom A(a, x) Hom C (F (a), c) a A Hom Set(Hom A (a, x), Hom C (F (a), c) = Nat ( HomA ( x), Hom C (F ( ), c ) Yoneda = HomC (F (x), c) natural in x A. The first line used that a coend in the contravariant argument of a Hom gives an end. Then we used the universal property of coproducts, followed by the adjunction from Observation Then, the expression of natural transformations as an end and finally in the last line the (contravariant) Yoneda lemma was used. The claim now follows by invoking the Yoneda lemma once again. The universal dinatural transformation consists of morphisms from Hom A (a, x) F (a) to F (x), for a A. Using the isomorphism Hom C (Hom A (a, x) F (a), F (x)) = Hom Set (Hom A (a, x), Hom C (F (a), F (x))) this morphism comes from the map f F (f) for all f Hom A (a, x). Example In particular, we find for any simplicial set X Fun(Δ opp, Set) X n Δ opp = Hom Δ opp([n], ) X([n]) n = Δ n X n where we used Hom Δ opp([n], ) = Hom Δ (, [n]) = Δ n sset. In this way, any simplicial set decomposes into standard simplices. 35

38 We recall the Yoneda completion ι : Γ ˆΓ := Fun(Γ opp, Set) γ Hom Γ (, γ) where the second category is cocomplete by remark Note that ι : Γ ˆΓ induces for D a cocomplete category an equivalence Fun c (ˆΓ, D) = Fun(Γ, D), where Fun c denote the cocontinuous functors. Indeed, we find Fun c (ˆΓ, D) def = Fun c (Fun(Γ opp, Set), D) = Fun c (Set, Fun(Γ, D)) = Fun(Γ, D) where we used that for any cocomplete category D we have Fun c (Set, D) = D by remark and an adjunction property of functor categories. This cocontinuous continuation for H : Γ D can be shown to be γ H(X) := H(γ) X(γ). We recognize in particular geometric realization as a cocontinous continuation of the functor Δ Top to sset, the Yoneda completion of Δ. Proposition With this notation, the functor H : ˆΓ D is left adjoint to the functor D ˆΓ T Hom D (H( ), T ) Fun(Γ, Set). Proof. For any X ˆΓ = Fun(Δ opp, Γ) and T D, we find Hom D ( H(X), T ) = γ Hom D(H(γ) X(γ), T ) = γ Hom Set(X(γ), Hom D (H(γ), T )) = Nat(X, Hom D (H( ), T )). where we used first the adjunction and then the description of natural transformations as an end. Theorem The singular complex functor S : Top sset is a right adjoint to the geometric realization : Top sset, i.e. there is a natural isomorphism: Hom Top ( K, X) = Hom sset (K, S (X)). In particular, the geometric realization functor is cocontinuous and the simplicial set functor is continuous. 36

39 Proof. We apply the preceding proposition and find for the right adjoint Top sset T ([n] Hom Top ( Δ n, T )) which is just the simplicial set functor. An explicit proof of this adjunction can also be found in [May, Chapter 15]. Corollary The singular simplicial set S(T ) = (Hom Top ( Δ n, T ) n N of a topological space T obeys the horn filling condition for any horn and is thus a Kan complex. Proof. Since geometric realization has by theorem a right adjoint, it is cocontinuous and thus preserves coequalizers. Thus the diagram Δ n 2 0i<jn n Δ n 1 Δ n, i=0 together with the fact that the geometric realization of Δ n is the standard simplex, yields the coequalizer diagram of topological spaces 0i<jn Δ n 2 n Δ n 1 Δ n. i=0 The induced map Δ n Δ n maps Δ n to the (n 1)-sphere bounding Δ n. Similarly, by remark horns are coequalizers. Thus Λ n k is the subspace of the topological simplex Δ n consisting of all faces except for the face opposite the k-th vertex. A horn on S(T ) corresponds to a continuous map Λ n k T. Now note that Λn k is a retract of Δ n and thus any continuous map Λ n k T can be extended, by precomposition with the retraction, to a continuous map Δ n T. One should appreciate that this extension is not unique. Now apply the adjunction to get a horn filler in the category of simplicial sets. Theorem The geometric realization is functor to CW, the subcategory of CW-complexes in Top. Proof. We describe the geometric realization inductively. Let X be a simplicial set. Define the n- skeleton sk n X of X as the subcomplex generated by simplices of X in degree n. Then X is a union X = n0 sk n X of the skeleta. Denote by NX n X n the set of non-degenerate simplices of X of degree n; these are those simplices x X n not of the form s i (y) with 0 i n 1 with y X n 1. There are pushout 37

40 diagrams x NXn Δ n sk n 1 X x NXn Δ n sk n X of simplicial sets. Thus X is a filtered colimit of topological spaces sk n X, where sk n X is obtained from sk n 1 X by attaching n-cells according to the pushout diagram x NXn Δ n sk n 1 X x NXn Δ n sk n X since geometric realization as a cocontinuous functor preserves colimits and thus pushouts. Note that we glue the spaces X n Δ n where X n is endowed with the discrete topology. Thus, we can think of X n Δ n as a disjoint union of cells. Thus X is a CW-complex. Remark One can show that for every CW-complex X, there is a homotopy equivalence X S (X) of topological spaces. (A homotopy equivalence is a map f : X Y having a homotopyinverse g : Y X, so f g id Y and g f id X.) We will define homotopies for Kan complexes in the next section. Then it makes sense to formulate that there is also a natural homotopy equivalence K S ( K ) for every Kan complex K. Thus geometric realization and taking singular complexes are inverses up to homotopy. For a full exposition, we refer to [May, Chapter 16] and, in a more modern treatment, [Goerss-Jardine, I.11]. As another instance where simplicial sets can capture all topological information, we mention: Remark Let Y be a topological space and U = (U α ) α I an open covering of Y. Consider the sets X n := {(α 0,..., α n ) U α0... U αn } I n+1 for n N and for a morphism f : [m] [n] in Δ the map X(f) : X n X m (α 0,..., α n ) (α f(0)..., α f(m) ) This is a simplicial set capturing the combinatorial structure of the covering. One can show that if the covering U is locally finite and if all non-empty finite intersections U α0... U αn are contractible, then the geometric realization X is homotopy equivalent to Y, encoding topology into combinatorial data. 38

41 We finally note that we can express left Kan extensions in terms of coends: Proposition Let C be a cocomplete category and f : Γ Γ be a functor of small categories. Consider the precomposition functor: f : Fun(Γ, C) Fun(Γ, C) G G f Then the functor is a left adjoint to f. Fun(Γ, C) Fun(Γ, C) F L f (F ) = γ Γ Hom Γ (f(γ), ) F (γ) Left adjoint functors are unique up to isomorphism. The left adjoint functor of precomposition is by Theorem a left Kan extension. Thus we have expressed the Kan extension in terms of a coend, γ Γ Lan f (F ) = Hom Γ (f(γ), ) F (γ) Similarly, a right Kan extension can be expressed in terms of an end. Proof. We refer to the proof of Proposition Explicitly, we calculate Nat(L f (F ), G) = Hom(Hom(f(γ), ) F (γ), G( )) [Hom is continuous] γ = Hom(Hom(f(γ), ), Hom(F (γ), G( )) [adjunction] γ Yoneda = Hom(F (γ), G(f(γ))) γ = Hom(F (γ), f G(γ)) γ = Nat(F, f G) Example We explore another instance of our construction: The category Δ of finite ordinals fully embeds into the category of categories by sending the totally ordered set [n] = {0,..., n} to the category with non-identity arrows given by the total order. On the other hand, we have again the Yoneda embedding Δ sset = Fun(Δ opp, Set). We wish to understand the left Kan extension Δ sset h Cat which associates to a simplicial set a category. By proposition , we know its right adjoint Cat sset C (n Hom sset (Δ n, C) = Hom Δ ([n], C)) 39

42 where we used the full embedding Δ sset. Thus we associate to a category the simplicial set which has in degree n strings of n composable arrows. Thus the right adjoint is the nerve functor from example We learn that, as a right adjoint functor, the nerve functor is continuous. The left adjoint h : sset Cat maps a simplicial set to its homotopy category. Interpreting the coend formula from proposition in Cat n Δ n Δ h(x) = Hom sset([n], X) [n] = X(n) [n] yields the following explicit description of the category hx associated to a simplicial set X. Its objects are represented by the 0-simplices, the vertices of X. Its arrows are freely generated from the 1-simplices: any composable path of 1-simplices oriented in correct direction represents a morphism in the category hx. They have to be subject to relations witnessed by 2-simplices: if there is a 2-simplex in X with 0-th face k, first face l, and second face j, then l = kj in hx. We will soon see a simpler description of this functor on the subcategory of -categories. 2.2 Simplicial homotopy In this subsection, we present a low tech approach to simplicial homotopy. We need a replacement for the interval. Definition We denote by I = Δ 1 the simplicial set with 2 non-degenerate 0-simplicies (0) and (1) and one non-degenerate 1-simplex (0, 1). 2. A path in a simplicial set X is a simplicial morphism p: I Xwhich by the Yoneda lemma amounts to a 1-simplex p X If p is a path in X, d 1 p = p(0) is called the initial point of the path and d 0 p = p(1) is called the final point or terminal point. 4. Two 0-simplices a and b of the simplicial set X are said to be in the same path component of X, if there is a path p with initial point a and final point b. Note that being in the same path component is, for a general simplicial set, not an equivalence relation. Consider, e.g., an ordered simplicial complex, e.g. in Δ n, in which we can have a < b or b < a but not both. The situation is different for Kan complexes: Proposition If X is a Kan complex, then being in the same path component is an equivalence relation. Proof. Reflexivity: for any vertex (a), s 0 (a) = (a, a) is a path from a to a. Transitivity: If p 1 is a path from a to b and p 2 is a path from b to c, then define a horn f : Λ 2 1 X mapping (0, 1) to p 1 and (1, 2) to p 2. The Kan condition guarantees and extension of f to ˉf : Δ 2 X, and ˉf(0, 2) gives us a path from a to c. Note that here only the extension for an inner horns is used. 40

43 Symmetry: Let p be a path in X from a to b. We need a path from b to a. To this end, we define a horn f : Λ 2 0 X mapping (0, 1) to p and (0, 2) to the constant path s 0 (a). The Kan for the outer horn condition implies that this map can be extended to all of Δ 2 and the image of (1, 2) gives a path from b to a. We present two classical versions of the definition of simplicial homotopy: one has the expected form for a homotopy, H : X I Y, the other is more closely related to the homotopies we see in chain complexes Ĥ : X n Y n+1. Definition [Simplicial homotopy 1] Two simplicial maps f, g : X Y are homotopic, if there is a simplicial map H : X I Y such that H X 0 = g and H X 1 = f, i.e. if g = H i 0 and f = H i 1, where i 0, i 1 are the evident simplicial inclusion maps i 0 : X [0] X I and i 1 : X [1] X I. Remark Notice that homotopies H : X I Y correspond to elements of the set Hom sset (X, Y ) 1 as defined in observation Recall that Elements of Hom sset (X, Y ) 0 correspond to simplicial maps. 2. If Y is a Kan complex, the simplicial set Hom sset (X, Y ) will be a Kan complex [May, Theorem 1.6.9]). Thus, in this case, two simplicial maps f, g Hom sset (X, Y ) 0 are homotopic, if and only if they lie in the same path component of Hom sset (X, Y ). Put differently, f, g : X Y are homotopic, if and only if they represent the same element of π 0 (Hom sset (X, Y )). The other definition is Definition [Simplicial homotopy 2] Two simplicial maps f, g : X Y are homotopic if for each 0 i p, there exist functions h i = h p i : X p Y p+1 such that 1. d 0 h 0 = f and d p+1 h p = g 2. d i h j = h j 1 d i if i < j d j+1 h j+1 = d j+1 h j d i h j = h j d i 1 if i > j s i h j = h j+1 s i if i j and s i h j = h j s i 1 if i > j. Note that the first condition is a shorthand for d p+1 0 h p 0 = f p and d p+1 p+1h p p = g p. Looking at the Moore complex of the corresponding freely generated abelian groups as in remark , we obtain differentials as alternating sums of simplicial maps p := p ( 1) i d i : i=0 41 ZX p ZX p 1

44 and maps h p := It is a direct exercise to check that p ( 1) i h p i : ZX p ZY p+1 i=0 p+1 h p + h p 1 p = f p g p. Indeed, p+1 h p + h p 1 p = p+1 r=0 s=0 p ( 1) r+s d r h s + p 1 p ( 1) r+s h s d r r =0 s =0 ( ) The first sum is decomposed and the relations are applied to find d 0 h 0 d p+1 h p + h s 1 d r ( 1) r+s + h s d r 1 ( 1) r+s r<s r>s+1 where the last two terms just cancel against the second sum in ( ). Proposition The two definitions and of homotopic maps coincide. Proof. We restrict ourselves to the case X = Δ p. Once we understand the case of a single simplex, everything else is determined by how the simplices of X are glued together. The key here is to recall how the prism Δ p I over Δ p is decomposed into p + 1 nondegenerate (p + 1)-simplices P k (Δ p I) p+1, 0 k p. Explicitly, we describe the simplices as follows: undashed indices are on the bottom lid of the prism, dashed indices on the top lid. Then the (p + 1)-simplices into which the simplicial prism Δ p I is decomposed are P k = [0,..., k, k,..., p ] with 0 k p, Δ p I. A homotopy H : Δ p I Y from f to g is determined by its restrictions on the (n + 1)- simplices P k, since every other nondegenerate simplex in Δ p I is a face of one of the simplices P k. To relate definition to definition definition 2.2.5, denote again the unique nondegenerate p-simplex of Δ p by ι p. In definition2.2.5, there are p + 1 maps h i : ι p Y p+1, each of which assigns to ι p a (p + 1)-simplex on Y. These maps together give a map from the prism over X to Y. 42

45 Given a homotopy H : Δ p I Y, introduce the map h k (ι p ) as the image H(P k ) in Y. We relate these maps to the conditions in definition and see what they mean. We discuss the first conditions, using the first definition of a homotopy: d 0 h 0 (ι p ) = d 0 H(P 0 ) [definition of h 0 ] = H(d 0 P 0 ) [H is a simplicial map] = H(d 0 [0, 0,..., p ]) [definition of P 0 ] = H([0,..., p ]) = H i 1 (ι p ) = f(ι p ) where in the end, we used the first definition. Similarly, d p+1 h p (ι p ) = H([0,..., p]) = H i 0 (ι p ) = g(ι p ). Thus the first conditions assure that the lids of the prism are controlled by the maps f and g. Most of the boundaries of the (p+1)-simplices on the prism Δ p I are themselves simplices of the prisms built on the boundary faces of X = Δ p. The first and third equations of the second set of conditions in definition ensure that these faces of the h i (Δ p ) are compatible with the actions of the homotopy maps h j i of lower dimensions j < p on the faces of Δ p. The second equation in the second set of equations, d j+1 h j+1 = d j+1 h j, is the condition that the neighboring simplices P k and P k+1 share a boundary. The third set equations implements the idea that the homotopy acts on degenerate simplices in a way that is determined by how it acts on the simplices of which they are degeneracies. For example, we have for i j that the i-th degeneracy of the (n + 1)-simplex P j on the prism is s i P j = [0,..., i, i,..., j, j,..., p ]. On the other hand, consider the the prism over the generate simplex [0,..., i, i,..., j 1, j, j + 1,..., p] of Δ p ; it has as the the (j + 1)st prism simplex just s i P j. Thus the ith degeneracy of the jth prism (p + 1)-simplex over Δ p is the (j + 1)st prism simplex over the ith degeneracy of Δ p, which explains the relation s i h j = h j+1 s i. One has to verify that conversely, the data given in definition suffice to construct a map H as in definition Remarks Path connectedness can now be rephrased by the statement that two maps Δ 0 X are homotopic. In proposition 2.2.2, we have seen that for Kan complexes, this is an equivalence relation. 2. More generally, homotopy of maps X Y is an equivalence relation, if Y is a Kan complex. If f and g are homotopic, we write f g. 43

46 3. One can also verify other expected elementary facts about homotopy; for instance if f f, then fg f g and gf gf. Also, homotopic maps between simplicial sets induce the same homomorphisms on homology groups as defined in remark , since they induce homotopies of chain complexes, by the remark after definition This is shown by an argument generalizing the usual proof in singular homology theory by using the triangulation of the homotopy prism, see [May, Section I.5]. We now want to define homotopy groups for Kan complexes. There are, at least, four different possibilities: 1. as homotopy classes of maps of (simplicial) spheres, cf. example to X. To this end, we need to fix some model of a simplicial sphere. 2. as the topological homotopy groups of the geometric realization of X, i.e. π n ( X, ). 3. As in algebraic topology, one can first define appropriate iterated simplicial loop spaces Ω n (X) and define π n (X) = π 0 (Ω n (X)). The path-space of a pointed Kan complex (K, ) is the simplicial set P (K, ) n := {x K n+1 d 0... d n x = } with the same face operators d P i := d i and degeneracy operators s P i := s i. The face map d n+1, which is not used as a face map of P, induces a simplicial map: d n+1 : P (K, ) K. The preimage of the base-point is the loop-space Ω(K, ) := d 1 n+1( ). 4. Using the following definition. For the equivalence of the definition we present and 1., we refer to [Friedman, Section 9]. We start by setting up a notion of homotopic simplices in a Kan complex. Observation Let K be a simplicial set and x, y K n two n-simplices with the same boundary, x = y. We want to use the two n-simplices to get an n-cycle which has these two simplices and otherwise just degeneracies: (s n 1 d 0 x,..., s n 1 d n 1 x, x, y) (K n ) n+2 We claim that this is an n-cycle. By remark 1.2.6, we have to check that d i x j = d j 1 x i for all 0 i < j n + 1. Indeed, for i = n, j = n + 1, we have d n y = d n x by the assumption x = y. For j = n + 1 and i < n, we have d n s n 1 d i x = d n s n 1 d i y = d i y, where we first used the assumption d i x = d i y and then the simplicial relation d n s n 1 = id. All other equtions follow similarly from the simplicial relations (1). Definition Two n-simplices x, y K n in a Kan complex K with the same boundary x = y are called homotopic, x y, if the n-cycle from observation is filled, i.e. if it is the boundary of an (n + 1)-simplex. 2. A filling (n + 1)-simplex is called a homotopy. We write h : x y. 44

47 We will use repeatedly the following Lemma (key-lemma). Let c = (x 0,..., x k 1, y, x k+1,..., x n ) be an (n 1)-boundary in a Kan complex K. Let y K n 1 be an (n 1)-simplex with the same boundary as the simplex y that is part of c. (This means y = y, which implies that that c := (x 0,..., y,... ) is a cycle, since the cycle condition in remark involves only boundaries.) Then the new cycle c is a boundary, if and only if the simplices y and y are homotopic, y y. Thinking of the simplicial set K = S (X) as the simplicial complex of some topological space X, one direction of the statement should become intuitive: gluing the homotopy and the filling together to a new filling, as the following picture illustrates. The proof is based on the following technical facts: Definition Consider an (n+2) tuple (c 0,..., c n+1 ) of (n 1)-cycles c i = (x i 0,..., x i n). It is called compatible, if it satisfies x i j = x j+1 i for all 0 i < j n + 1. Explicitly, the matrix ((c 0 ) T,..., (c n+1 ) T ) with columns given by the cycles has the following shifted mirror symmetry: x 0 0 x 0 x 0 0 x 0 1 x 1 1 x 0 1 x 1 n 1 x 0 1 x 1 n x 0 n 1 x 1 n+1 n x 1 n x 0 n x 1 n x 2 n x n n x n n x n n+1 Lemma Suppose that (c 0,..., c n+1 ) is an (n + 2)-tuple of compatible cycles in a Kan complex. If all cycles except for the cycle c k for some 0 k n + 1 are boundaries, then the cycle c k has to be a boundary as well. Proof. Find n-simplices y i K n for all i k such that y i = c i. Then (y 0,..., y k 1,?, y k+1,..., y n+1 ) is a horn. Indeed, we have from y i = c i that d j y i = x i j. This in turn implies the horn condition d j 1 y i = x i j 1! = x j i = d iy j for i < j where at! we have used that the cycles are compatible. By the Kan property of K, we can complete this horn by some n-simplex y k K n. We compute the boundary y k and find d i y k = d k 1 y i = c k 1 i Thus y k = c k. Thus the cycle c k is a boundary.! = c i k for i < k and d i y k = d k y i+1 = c k i+1 = c i k for k i. 45